For years, Einhorn has decried Musk's controversial leadership style and called for him to be ousted from the CEO seat at Tesla. His October 30 letter to shareholders rails against Musk's handling of the SolarCity takeover, which Tesla acquired for $2.6 billion in 2016. SolarCity, which was run by Musk's cousin, has been the target numerous lawsuits and fraud allegations stretching back to before the acquisition.

One lawsuit filed by Tesla shareholders alleges Musk engineered the purchase of SolarCity, in which he was the majority shareholder, as a a bailout of the troubled solar panel company. That case is heading for trial in March.

Over the past two years, Tesla's electric car business has been beleaguered with manufacturing problems related to its rollout of the new Model 3 sedan. The company's latest earnings report signals the firm may be at a turning point. It posted a $342 million profit, indicating it's making bigger margins on car sales.

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Musk also invited Einhorn to tour the Tesla factory and learn more about the company, adding "I'm certain your investors would appreciate you getting smart on Tesla."

Einhorn responded Friday, in an open letter challenging Musk to identify inaccuracies in Greenlight's shareholder letter and accepting the invitation to visit Tesla factories, writing, "I think facility visits would be fun (can we start in Buffalo?). I might learn the difference between your alien dreadnought factory and cars made by hand in a tent."